We prove an exponential lower bound for tree-like Cutting Planesrefutations of a set of clauses which has polynomial size resolutionrefutations. This implies an exponential separation between tree-likeand dag-like proofs for both Cutting Planes and resolution; in bothcases only superpolynomial separations were known before.In order to prove this, we extend the lower bounds on the depth ofmonotone circuits of Raz and McKenzie (FOCS 97) to monotone realcircuits.

In the case of resolution, we further improve this result by giving anexponential separation of tree-like resolution from (dag-like) regularresolution proofs. In fact, the refutation provided to give the upperbound respects the stronger restriction of being a Davis-Putnamresolution proof. This extends the corresponding superpolynomialseparation of Urquhart (BSL 1995).

Finally, we prove an exponential separation between Davis-Putnamresolution and unrestricted resolution proofs; only a superpolynomialseparation was previously known (Goerdt, Ann. Math. AI 1992).